(Ex 9915) lrc circuit
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This document describes an experiment involving measuring voltages and currents in various AC circuits containing resistors, capacitors, and inductors. Key points: 1) An LC circuit is used to measure the resonant frequency and calculate the inductance. Current and voltage relationships are examined for resistive, capacitive, and inductive circuits individually. 2) Current and voltage measurements are taken for an LRC circuit as the frequency is varied to observe the resonance curve. Peak current frequency agrees with theoretical LC resonance frequency. 3) Voltage sensors are added to an LRC circuit to measure voltages across each component and verify Kirchhoff's loop rule and theoretical phase relationships.
(Ex 9915) lrc circuit
- 1. LRC Circuit EX-9915 Page 1 of 10 1LRC CIRCUIT EQUIPMENT INCLUDED: 1 AC/DC Electronics Laboratory EM-8656 3 Voltage Sensors CI-6503 NOT INCLUDED, BUT REQUIRED: 1 ScienceWorkshop 750 Interface CI-7650 1 DataStudio Software CI-6870 INTRODUCTION A square wave voltage is applied to an LC circuit and the period of oscillation of the voltage across the capacitor is measured and compared to the theoretical value. Then three AC circuits are examined: A sinusoidal voltage is applied individually to a resistor, a capacitor, and an inductor. The amplitude of the current and the phase difference between the applied voltage and the current are measured in each of the three circuits to see the effect each component has on the current. Finally, a sinusoidal voltage is applied to an inductor, resistor, and capacitor in series. The amplitude of the current and the phase difference between the applied voltage and the current are measured and compared to theory. THEORY LC Oscillations A low frequency square wave voltage is applied to an inductor and capacitor series circuit. This charges the capacitor. Then the capacitor discharges through the inductor and the voltage across the capacitor oscillates at the resonant frequency of the LC circuit, 1 ω= . (1) LC Resistive Circuit A resistive circuit is one in which the dominant component is resistance. In this experiment, a voltage V = Vmaxsin(ωt) is applied to a resistor alone. The current, I, through the resistor is in phase with the applied voltage: I = Imaxsin(ωt) (2) where the maximum current, Imax, is equal to Vmax/R. (3) Written by Ann Hanks
- 2. LRC Circuit EX-9915 Page 2 of 10 Capacitive Circuit A capacitive circuit is one in which the dominant component is capacitance. In this part of the experiment, a voltage V = V maxsin(ωt) is applied to a capacitor alone. Since the voltage across the capacitor is Q V = = V max sin(ωt ) C where Q is the charge on the capacitor and C is the capacitance. Solving for Q and using dQ I = , the phase difference between the current through the capacitor and the applied voltage dt is π/2: π I = I max cos ωt = I max sinωt + (4) 2 Vmax 1 where I = and the capacitive reactance is χC = . (5) χC ωC Inductive Circuit An inductive circuit is one in which the dominant component is inductance. In this part of the experiment, a voltage V = V maxsin(ωt) is applied to an inductor alone. Note that the resistance of the inductor is ignored in the theory. The voltage across the inductor is dI V =L = Vmax sin(ωt ) dt where I is the current through the inductor and L is the inductance. Solving for I using I = Ι( Vmax/L)sin(ωt)dt, π I = −I max cos ωt = I max sin ωt − (6) 2 Vmax where I max = and the inductive reactance is χ L = ωL . (7) χL The phase difference between the current through the inductor and the applied voltage is -π/2. LRC Circuit A voltage V = V maxsin(ωt) is applied to an inductor, capacitor, and resistor in series. The resulting current is given by I = Imaxsin(ωt + φ) (8) ξmax where I max = and Z is the impedance, Z = R 2 + ( X C − X L ) 2 . (9) Z The phase constant (φ) is the phase difference between the current and applied voltage. Note that with the sign convention used here, φ is positive when the current leads the emf (V). XC − XL tan φ = (10) R Written by Ann Hanks
- 3. LRC Circuit EX-9915 Page 3 of 10 LC Oscillations SETUP 1. Connect the output voltage to the input banana jacks on the AC/DC circuit board using banana plugs. Connect the inductor and 10µF capacitor in series with the applied voltage as shown in Figure 1. IMPORTANT: Insert the iron core into the inductor. L Applied C Voltage Figure 1: LC Circuit 2. Open the DataStudio file called "LC Circuit". Set the 750 signal generator on a 3-Volt square wave having a frequency of 30 Hz. The 750 interface automatically measures the applied voltage and the resulting current. No Voltage Sensors are needed for this part of the experiment. PROCEDURE 1. Click on START. Adjust the time axis on the oscilloscope so that about one cycle of the square wave is visible. Click STOP. 2. For more exact measurements, click on the Applied Voltage box on the right of the oscilloscope and click on the data export button at the top of the oscilloscope. This will make a data file for the voltage. 3. Click on the Current box on the right of the oscilloscope and click on the data export button at the top of the oscilloscope. This will make a data file for the current. 4. Click on the data graph and use the Smart Cursor to measure the period of the LC oscillation. Use f = 1/T to find the frequency. 5. Using the theoretical resonant frequency in Equation (1) and assuming the capacitance is its stated value, calculate the inductance of the inductor with the core. Note that the equation has the angular frequency but the linear frequency was found from the graph. π Convert the angular frequency to linear frequency using f = . 2 Written by Ann Hanks
- 4. LRC Circuit EX-9915 Page 4 of 10 Resistive Circuit SETUP 1. Change the circuit to a 10 Ω resistor in series with the applied voltage as shown in Figure 2. Applied R Voltage C Figure 2: Resistive Circuit 2. Open the DataStudio file called "R Circuit". Set the 750 signal generator on a 3-Volt sine wave having a frequency of 100 Hz. The 750 interface automatically measures the applied voltage and the resulting current. No Voltage Sensors are needed for this part of the experiment. PROCEDURE 1. Click on START. Adjust the time axis on the oscilloscope so that one or two cycles of the wave are visible. Click STOP. 2. Use the Smart Cursor to examine the phase difference between the applied voltage and the current. The phase difference is the angle between the peaks of the two sine waves. It is calculated using ∆T ∆T φ= x360 o or φ= x 2π radians. (11) T T ΔT is the time between the peak of one wave and the peak of the other wave. T is the period of the waves which is the inverse of the applied frequency. 3. What is the phase difference between the resistor voltage and the current? 4. Increase the frequency of the applied voltage to 1000 Hz. Does the phase difference change? Does the magnitude of the current change? Written by Ann Hanks
- 5. LRC Circuit EX-9915 Page 5 of 10 Capacitive Circuit SETUP 1. Change the circuit to a 10 µF capacitor in series with the applied voltage as shown in Figure 3. Applied C Voltage C Figure 3: Capacitive Circuit 2. Open the DataStudio file called "C Circuit". Set the 750 signal generator on a 3-Volt sine wave having a frequency of 100 Hz. The 750 interface automatically measures the applied voltage and the resulting current. No Voltage Sensors are needed for this part of the experiment. PROCEDURE 1. Click on START. Adjust the time axis on the oscilloscope so that one or two cycles of the wave are visible. Click STOP. 2. Use the Smart Cursor to examine the phase difference between the applied voltage and the current. Calculate the phase difference using Equation (11). 3. What is the phase difference between the capacitor voltage and the current? 4. Increase the frequency of the applied voltage to 1000 Hz. Does the phase difference change? Does the magnitude of the current change? Inductive Circuit SETUP 1. Change the circuit to the inductor with L the iron core in the coil in series with the applied voltage as shown in Figure 4. Applied C Voltage Figure 4: Inductive Circuit Written by Ann Hanks
- 6. LRC Circuit EX-9915 Page 6 of 10 2. Open the DataStudio file called "L Circuit". Set the 750 signal generator on a 3-Volt sine wave having a frequency of 1000 Hz. The 750 interface automatically measures the applied voltage and the resulting current. No Voltage Sensors are needed for this part of the experiment. PROCEDURE 1. Click on START. Adjust the time axis on the oscilloscope so that one or two cycles of the wave are visible. Click STOP. 2. Use the Smart Cursor to examine the phase difference between the applied voltage and the current. Calculate the phase difference using Equation (11). 3. What is the phase difference between the inductor voltage and the current? 4. Decrease the frequency of the applied voltage to 100 Hz. Does the phase difference change? Does the magnitude of the current change? LRC Circuit SETUP 1. Change the circuit to the inductor with the iron core in the coil, the 10 µF capacitor, and the 10 Ω resistor in series with the applied voltage as shown in Figure 5. L C C Applied R Voltage C Figure 5: LRC Circuit without Voltage Sensors 2. Open the DataStudio file called "Resonance Curve". Set the 750 signal generator on a 3- Volt sine wave having a frequency of 20 Hz. The 750 interface automatically measures the applied voltage and the resulting current. No Voltage Sensors are needed for this part of the experiment. Written by Ann Hanks
- 7. LRC Circuit EX-9915 Page 7 of 10 PROCEDURE 1. Click on START. Adjust the time axis on the oscilloscope so that one or two cycles of the wave are visible. Allow the oscilloscope to run throughout this part of the experiment. 2. Click on the Current box on the right of the oscilloscope and then use the Smart Cursor to measure the maximum current. Enter the signal generator frequency and the maximum current into the table labeled "10 Ohm". 3. Increase the signal generator frequency by 20 Hz and find the new current and record the results in the table. Continue to increase the frequency by increments you choose by watching the resulting resonance graph. If necessary, go back and take data points at lower frequencies to fill in the resonance curve. If you go back, insert new rows into the table at the appropriate points so the connecting line on the graph will connect the points in order of frequency. 4. Continue until you reach 3000 Hz. Be sure you get enough data points around the peak. 5. Determine the frequency corresponding to the peak in the maximum current. Compare this frequency to the theoretical resonance frequency given by Equation (1). 6. Measure the phase difference between the current and the applied voltage at a frequency of 100 Hz, at 3000 Hz, and at the resonant frequency. 7. Replace the 10 Ω resistor with a 100 Ω resistor and repeat Steps 2 through 4. Take a smaller number of data points for this resistor, concentrating on the lower frequencies, the resonant frequency, and the higher frequencies so the general shape of the curve is distinguished. QUESTIONS 1. Does the frequency of the peak maximum current correspond to the frequency of the LC oscillations? Does changing the resistance change the frequency of the peak? 2. Using Equation (9), calculate the theoretical peak current at resonance. Why is the actual peak current at resonance less than the theoretical? 3. Why is the peak lower for the greater resistance? 4. Explain why the phase shifts are what they are at low frequency and at high frequency and at the resonant frequency. In each case, does the current lead or lag the applied voltage? To what value does the phase shift go as the frequency goes to zero or as it goes to infinity? Which components dominate the circuit at these different frequencies? Written by Ann Hanks
- 8. LRC Circuit EX-9915 Page 8 of 10 5. Draw phasor diagrams (approximately to scale) for 100 Hz, 3000 Hz, and the resonant frequency. LRC VOLTAGES SETUP 1. Change the circuit to the inductor with the iron core in the coil, the 10 μF capacitor, and the 10 Ω resistor in series with the applied voltage as shown in Figure 6. Add 3 Voltage Sensors to the circuit: One across the inductor plugged into Channel A, one across the capacitor plugged into Channel B, and one across the resistor plugged into Channel C. Be careful to plug the sensors into the assigned channels on the interface and pay attention to the polarity of each sensor. Voltage Sensor V L L C V Voltage Sensor C C Applied Voltage Voltage R V Sensor C R Figure 6: LRC Circuit with Voltage Sensors 2. Open the DataStudio file called "LRC Voltages". Set the 750 signal generator on a 3- Volt sine wave having a frequency of 200 Hz. The 750 interface automatically measures the applied voltage and the resulting current. Written by Ann Hanks
- 9. LRC Circuit EX-9915 Page 9 of 10 PROCEDURE 1. Click on START. Adjust the time axis on the oscilloscope so that one cycle of the waves is visible. Click on STOP. Note that all the voltage scales are set at 1V/division so the amplitudes of the voltages can be easily compared. 2. To compare the voltages in greater detail, the oscilloscope data must be exported to a graph. To do this, click on the "Voltage across L" box on the right of the oscilloscope and click on the data export button at the top of the oscilloscope. This will make a data file for the voltage. Repeat this procedure for each of the boxes on the right of the oscilloscope. 3. Drag each of the voltage data in the Data list at the left to a single graph. 4. Click the Lock button at the top of the graph to lock all the time axes together. 5. One by one, click on each of the Legends on the graph and then click on the Smart Cursor at the top of the graph to turn on a cursor for each wave. Move the cursors so they seek the data points in each wave and are locked into each other. 6. Place the cursor on the Applied Voltage on a point where the wave crosses the voltage axis. For this particular time, read the voltages across the inductor, capacitor, and resistor. Algebraically add these three voltages together. Does the sum equal the Applied Voltage at that time? Does Kirchhoff's Loop Rule apply to AC circuits? 7. Using the cursors, determine the phase difference between the Applied Voltage and the Current. Calculate the theoretical phase shift using Equation (10). Calculate the percent difference between the measured and theoretical values. 8. Calculate the impedance at this frequency using Equation (9) and use it to calculate the theoretical maximum current. Measure the maximum current on the graph and compare. NOTE: If an ohmmeter is available, the resistance of the inductor can be measured and included in the total resistance of the circuit for these calculations. 9. Measure the phase differences between the Applied Voltage and the voltages across each component in the circuit (L, C, and R). Compare to theory. QUESTIONS 1. How does the resistance of the inductor affect the phase difference between the current and the applied voltage? Draw a phasor diagram which includes the fact that the voltage across the inductor includes the resistance of the inductor. 2. Why is the measured phase difference for the capacitor and applied voltage the same as theory while the phase for the inductor is not in agreement with theory? Written by Ann Hanks
- 10. LRC Circuit EX-9915 Page 10 of 10 3. Why is the phase difference between the current and the resistor voltage zero? 4. Is this circuit being driven at resonance, or below resonance, or above resonance? How do you know? Give at least two reasons. Written by Ann Hanks
- 11. LRC Circuit EX-9915 Page 10 of 10 3. Why is the phase difference between the current and the resistor voltage zero? 4. Is this circuit being driven at resonance, or below resonance, or above resonance? How do you know? Give at least two reasons. Written by Ann Hanks
- 12. LRC Circuit EX-9915 Page 10 of 10 3. Why is the phase difference between the current and the resistor voltage zero? 4. Is this circuit being driven at resonance, or below resonance, or above resonance? How do you know? Give at least two reasons. Written by Ann Hanks